A blog is that is all about mathematics and calculators, two of my passions in life.

Sunday, June 29, 2014

Bow Ties and Cylindrical Hourglasses

The Area of a
Bow Tie

Consider the
following bow-tie with the following dimensions:

A = length of
the large (outer) line

B = length of
the smaller (inner) line

H = half of
the height (or length) of the bow tie

The bow tie
is two trapezoids stacked on each other, with their narrow ends meeting each
other.Hence, the area of the bow tie is
twice the area of a trapezoid.

The area of a
trapezoid is:

A = 2 *(area
of the trapezoid)

A = 2 *(1/2 *
H * (A + B))

A = H * (A +
B)

Fairly
easy.

The Volume of
a Cylindrical Hourglass

I do not
think there is an official name for this, but I am naming it the cylindrical hourglass
because it is built by two identical cylindrical frustums, with their smaller
bases attached. Hence the volume of the hourglass is twice the volume of one of
the cylindrical frustum (frustum of a cylinder).

Let:a = A/2 = small (inner) radius,

b = B/2 =
larger (outer) radius, and

h = H =
height of half of the hourglass

We know the
volume of a frustum of a cylinder is:

V = h/3 * π *
(a^2 + a*b + b^2)

** For a
derivation of the volume of frustum of a cylinder, see the next section.

Since the
volume of the cylindrical hourglass is twice this volume, the final volume is:

V = 2 * h/3 *
π * (a^2 + a*b + b^2)

Volume of a
Frustum of a Cylinder

Imagine the
cross section of the frustum placed on a Cartesian plane.Let the bottom of the base “rest” on the
x-axis as seen on the above picture.We
will calculate the volume of the frustum using the Method of Discs.For this calculation, let the discs revolve
around the y-axis.The general formula
for volume with the discs revolving around the y-axis is:

1 comment:

Great topic and clear explanations Eddie! Integrating these types of fomulae opens up great opportunities. The HP 35s is a great tool for exploring integration on a budget. One can get a lot of mileage from this post. Thanks for sharing!

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Went to Cal Poly Pomona. Degrees: Accounting (Bachelor of Science), Mathematics (Masters of Science). My email is ews31415@gmail.com. I am also on Twitter: @edward_shore. I can also be found on Google+ under Edward Shore.